3.2.45 \(\int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^5} \, dx\) [145]

Optimal. Leaf size=64 \[ \frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[Out]

1/12*b*arccoth(tanh(b*x+a))^3/x^3/(b*x-arccoth(tanh(b*x+a)))^2+1/4*arccoth(tanh(b*x+a))^3/x^4/(b*x-arccoth(tan
h(b*x+a)))

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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2202, 2198} \begin {gather*} \frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^2/x^5,x]

[Out]

(b*ArcCoth[Tanh[a + b*x]]^3)/(12*x^3*(b*x - ArcCoth[Tanh[a + b*x]])^2) + ArcCoth[Tanh[a + b*x]]^3/(4*x^4*(b*x
- ArcCoth[Tanh[a + b*x]]))

Rule 2198

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-u^(m + 1))*(
v^(n + 1)/((m + 1)*(b*u - a*v))), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] &&
EqQ[m + n + 2, 0] && NeQ[m, -1]

Rule 2202

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-u^(m + 1))*(
v^(n + 1)/((m + 1)*(b*u - a*v))), x] + Dist[b*((m + n + 2)/((m + 1)*(b*u - a*v))), Int[u^(m + 1)*v^n, x], x] /
; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^5} \, dx &=\frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^4} \, dx}{4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.58 \begin {gather*} -\frac {b^2 x^2+2 b x \coth ^{-1}(\tanh (a+b x))+3 \coth ^{-1}(\tanh (a+b x))^2}{12 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Tanh[a + b*x]]^2/x^5,x]

[Out]

-1/12*(b^2*x^2 + 2*b*x*ArcCoth[Tanh[a + b*x]] + 3*ArcCoth[Tanh[a + b*x]]^2)/x^4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.34, size = 3217, normalized size = 50.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(3217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(tanh(b*x+a))^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4/x^4*ln(exp(b*x+a))^2-1/24*(4*b*x+3*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+
1))^2-6*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3+6*I*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-3*I*Pi*csgn(I*exp
(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+3*I*Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+6*
I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2-3*I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-3*I*Pi*csgn(I*exp(2*b*x+2
*a))^3-6*I*Pi-3*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
)))/x^4*ln(exp(b*x+a))-1/192*(12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))^3-12*Pi^2*csgn(I/(ex
p(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+16*I*Pi*b*x*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x
+2*a))^2+16*I*Pi*b*x*csgn(I/(exp(2*b*x+2*a)+1))^2-12*Pi^2-12*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*
csgn(I/(exp(2*b*x+2*a)+1))^3-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(
exp(2*b*x+2*a)+1))-16*I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^3+12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*
x+2*a)/(exp(2*b*x+2*a)+1))^2-12*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+24*Pi^2*csgn(I*exp(b*x+a))*cs
gn(I*exp(2*b*x+2*a))^2+12*Pi^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-8*I*Pi*b*x*c
sgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-6*Pi^2*csgn(I/(exp(
2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+6*Pi^2*
csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))^2+12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp
(2*b*x+2*a)+1))-16*I*Pi*x*b-6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*a)/(
exp(2*b*x+2*a)+1))+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x
+2*a)+1))^2+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))^3+12*Pi^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3-18*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))
^4+12*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+6*Pi^2*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*a)/(
exp(2*b*x+2*a)+1))^2-3*Pi^2*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*c
sgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4+6*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x
+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-6*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I
*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-12*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a
)/(exp(2*b*x+2*a)+1))^2+12*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2
*a)+1))^3-3*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))^2+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+16
*b^2*x^2+8*I*Pi*b*x*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+8*I*Pi*b*x*csgn(I/(exp(
2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-8*I*Pi*b*x*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*
a))-8*I*Pi*b*x*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-3*Pi^2*csgn(I*exp(2*b*x+2*a))^6+6*Pi^2*csgn(I/(exp(
2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5-6*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*
a)/(exp(2*b*x+2*a)+1))^3+6*Pi^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5-12*Pi^2*csg
n(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^
2-3*Pi^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4-3*Pi^2*csgn(I*exp(2*b*x+2*a)/(ex
p(2*b*x+2*a)+1))^6-3*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4-12*Pi^2*csg
n(I/(exp(2*b*x+2*a)+1))^6-12*csgn(I/(exp(2*b*x+2*a)+1))^4*Pi^2+24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^5+12*Pi^2*cs
gn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-12*Pi^2*csgn(I/(ex
p(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+24*Pi^2*csgn(I/(exp(2*b*
x+2*a)+1))^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-12*Pi^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1
2*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a))^3+12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2
*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3+12*Pi^2*csgn(I*exp
(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x+2*a)+1))^2-24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp
(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x
+2*a)/(exp(2*b*x+2*a)+1))+12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+24*
csgn(I/(exp(2*b*x+2*a)+1))^2*Pi^2-8*I*Pi*b*x*cs...

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Maxima [A]
time = 0.34, size = 36, normalized size = 0.56 \begin {gather*} -\frac {b^{2}}{12 \, x^{2}} - \frac {b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{6 \, x^{3}} - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="maxima")

[Out]

-1/12*b^2/x^2 - 1/6*b*arccoth(tanh(b*x + a))/x^3 - 1/4*arccoth(tanh(b*x + a))^2/x^4

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Fricas [A]
time = 0.34, size = 29, normalized size = 0.45 \begin {gather*} -\frac {24 \, b^{2} x^{2} + 32 \, a b x - 3 \, \pi ^{2} + 12 \, a^{2}}{48 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="fricas")

[Out]

-1/48*(24*b^2*x^2 + 32*a*b*x - 3*pi^2 + 12*a^2)/x^4

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Sympy [A]
time = 0.34, size = 39, normalized size = 0.61 \begin {gather*} - \frac {b^{2}}{12 x^{2}} - \frac {b \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{6 x^{3}} - \frac {\operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(tanh(b*x+a))**2/x**5,x)

[Out]

-b**2/(12*x**2) - b*acoth(tanh(a + b*x))/(6*x**3) - acoth(tanh(a + b*x))**2/(4*x**4)

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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 38, normalized size = 0.59 \begin {gather*} -\frac {24 \, b^{2} x^{2} + 16 i \, \pi b x + 32 \, a b x - 3 \, \pi ^{2} + 12 i \, \pi a + 12 \, a^{2}}{48 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="giac")

[Out]

-1/48*(24*b^2*x^2 + 16*I*pi*b*x + 32*a*b*x - 3*pi^2 + 12*I*pi*a + 12*a^2)/x^4

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Mupad [B]
time = 1.17, size = 36, normalized size = 0.56 \begin {gather*} -\frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4\,x^4}-\frac {b^2}{12\,x^2}-\frac {b\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{6\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(tanh(a + b*x))^2/x^5,x)

[Out]

- acoth(tanh(a + b*x))^2/(4*x^4) - b^2/(12*x^2) - (b*acoth(tanh(a + b*x)))/(6*x^3)

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